Problem: $ D = \left[\begin{array}{rrr}-1 & 4 & -2 \\ 0 & -1 & 1 \\ -2 & 4 & 2\end{array}\right]$ $ B = \left[\begin{array}{rr}4 & 3 \\ 2 & 4 \\ -2 & 4\end{array}\right]$ Is $ D B$ defined?
Answer: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ D$ , have? How many rows does the second matrix, $ B$ , have? Since $ D$ has the same number of columns (3) as $ B$ has rows (3), $ D B$ is defined.